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Current Mechanics and Advanced Materials

Author(s): Maziar Janghorban* and Behrouz Karami

DOI: 10.2174/2666184501999201005211608

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Free Vibration of Functionally Graded Carbon Nanotube-reinforced Doubly-curved Shells

Page: [39 - 49] Pages: 11

  • * (Excluding Mailing and Handling)

Abstract

Background: Carbon nanotubes (CNTs) reinforced structures are the main elements of structural equipment. Hence a wide range of investigations has been performed on the response of these structures. A lot of studies covered the static and dynamic phenomenon of CNTs reinforced beams, plates and shells. However, there is no study on the free vibration analysis of a doubly-curved nano-size shell made of CNTs reinforced composite materials.

Methods: This work utilized a general third-order shear deformation theory to model the nanoshell where the general strain gradient theory is used in order to capture both nonlocality and strain gradient size-dependency. The Navier solution solving procedure is adopted to solve the partial differential equations (PDEs) and get the natural frequency of the system which is obtained through the Hamilton principle.

Results: The current study shows the importance of small-scale coefficients. The natural frequency increases with rising the strain gradient-size dependency which is because of stiffness enhancement, while the natural frequency decreases by increasing the nonlocality. In addition, the numerical examples covered the CNTs distribution patterns.

Conclusion: This work also studied the importance of shell panel’s shape. It has been observed that spherical shell panel has a higher frequency compared to the hyperbolic one. Furthermore, the frequency of the system increases with growing length-to-thickness ration.

Keywords: Carbon nanotube-reinforced nanocomposites, doubly-curved shell, free vibration, higher-order shear deformation theory, nonlocal strain gradient theory, hamilton principle.

Graphical Abstract

[1]
D. Qian, E.C. Dickey, R. Andrews, and T. Rantell, "Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites", Appl. Phys. Lett., vol. 76, no. 20, pp. 2868-2870, 2000.
[http://dx.doi.org/10.1063/1.126500]
[2]
M. Motezaker, and A. Eyvazian, "Post-buckling analysis of Mindlin Cut out-plate reinforced by FG-CNTs", Steel Compos. Struct., vol. 34, no. 2, pp. 289-297, 2020.
[3]
B. Karami, D. Shahsavari, M. Janghorban, and L. Li, "Elastic guided waves in fully-clamped functionally graded carbon nanotube-reinforced composite plates", In: Mater. Res. Express, vol. 6. 2019, no. 9, p. 0950a9.
[http://dx.doi.org/10.1088/2053-1591/ab3474]
[4]
B. Karami, M. Janghorban, D. Shahsavari, R. Dimitri, and F. Tornabene, "Nonlocal buckling analysis of composite curved beams reinforced with functionally graded carbon nanotubes", Molecules, vol. 24, no. 15, p. 2750, 2019.
[http://dx.doi.org/10.3390/molecules24152750] [PMID: 31362407]
[5]
B. Karami, D. Shahsavari, and M. Janghorban, "A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates", Aerosp. Sci. Technol., vol. 82, pp. 499-512, 2018.
[http://dx.doi.org/10.1016/j.ast.2018.10.001]
[6]
A.C. Eringen, and D. Edelen, "On nonlocal elasticity", Int. J. Eng. Sci., vol. 10, no. 3, pp. 233-248, 1972.
[http://dx.doi.org/10.1016/0020-7225(72)90039-0]
[7]
S. Papargyri-Beskou, D. Polyzos, and D. Beskos, "Wave dispersion in gradient elastic solids and structures: a unified treatment", Int. J. Solids Struct., vol. 46, no. 21, pp. 3751-3759, 2009.
[http://dx.doi.org/10.1016/j.ijsolstr.2009.05.002]
[8]
F. Yang, A. Chong, D.C.C. Lam, and P. Tong, "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., vol. 39, no. 10, pp. 2731-2743, 2002.
[http://dx.doi.org/10.1016/S0020-7683(02)00152-X]
[9]
H. Askes, and E.C. Aifantis, "Gradient elasticity and flexural wave dispersion in carbon nanotubes", In: Phys. Rev. B, vol. 80. no. 19, . 2009.195412
[http://dx.doi.org/10.1103/PhysRevB.80.195412]
[10]
M.R. Barati, and H. Shahverdi, "Hygro-thermal vibration analysis of graded double-refined-nanoplate systems using hybrid nonlocal stress-strain gradient theory", Compos. Struct., vol. 176, pp. 982-995, 2017.
[http://dx.doi.org/10.1016/j.compstruct.2017.06.004]
[11]
B. Karami, M. Janghorban, and A. Tounsi, "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., vol. 27, no. 2, pp. 201-216, 2018.
[12]
A. Norouzzadeh, and R. Ansari, "Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity", Physica E, vol. 88, pp. 194-200, 2017.
[http://dx.doi.org/10.1016/j.physe.2017.01.006]
[13]
L. Li, and Y. Hu, "Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., vol. 120, pp. 159-170, 2017.
[http://dx.doi.org/10.1016/j.ijmecsci.2016.11.025]
[14]
L. Li, Y. Hu, and X. Li, "Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory", Int. J. Mech. Sci., vol. 115, pp. 135-144, 2016.
[http://dx.doi.org/10.1016/j.ijmecsci.2016.06.011]
[15]
L. Li, Y. Hu, and L. Ling, "Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory", Compos. Struct., vol. 133, pp. 1079-1092, 2015.
[http://dx.doi.org/10.1016/j.compstruct.2015.08.014]
[16]
B. Karami, D. Shahsavari, M. Janghorban, and L. Li, "Wave dispersion of mounted graphene with initial stress", Thin-walled Struct., vol. 122, pp. 102-111, 2018.
[http://dx.doi.org/10.1016/j.tws.2017.10.004]
[17]
A. Norouzzadeh, R. Ansari, and H. Rouhi, "An analytical study on wave propagation in functionally graded nano-beams/tubes based on the integral formulation of nonlocal elasticity", Waves Random Complex Media, vol. 30, no. 3, pp. 562-580, 2020.
[http://dx.doi.org/10.1080/17455030.2018.1543979]
[18]
S. Adali, "Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory", Phys. Lett. A, vol. 372, no. 35, pp. 5701-5705, 2008.
[http://dx.doi.org/10.1016/j.physleta.2008.07.003]
[19]
R. Ansari, M.F. Oskouie, and R. Gholami, "Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory", Physica E, vol. 75, pp. 266-271, 2016.
[http://dx.doi.org/10.1016/j.physe.2015.09.022]
[20]
E.C. Aifantis, "On the gradient approach-relation to Eringen’s nonlocal theory", Int. J. Eng. Sci., vol. 49, no. 12, pp. 1367-1377, 2011.
[http://dx.doi.org/10.1016/j.ijengsci.2011.03.016]
[21]
S. Ziaee, "Linear free vibration of graphene sheets with nanopore via Aifantis theory and Ritz method", J. Theor. Appl. Mech., vol. 55, no. 3, pp. 823-838, 2017.
[http://dx.doi.org/10.15632/jtam-pl.55.3.823]
[22]
B. Karami, and M. Janghorban, "On the mechanics of functionally graded nanoshells", Int. J. Eng. Sci., vol. 153, . 2020.103309
[http://dx.doi.org/10.1016/j.ijengsci.2020.103309]
[23]
B. Karami, M. Janghorban, and T. Rabczuk, "“Forced vibration analysis of functionally graded anisotropic nanoplates resting on Winkler/Pasternak-foundation”, CMC-Comp", Mater. Continua, vol. 62, no. 2, pp. 607-629, 2020.
[http://dx.doi.org/10.32604/cmc.2020.08032]
[24]
B. Karami, M. Janghorban, and T. Rabczuk, "Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory", In: Compos., Part B Eng., vol. 182. . 2020.107622
[http://dx.doi.org/10.1016/j.compositesb.2019.107622]
[25]
A. Anjomshoa, A.R. Shahidi, B. Hassani, and E. Jomehzadeh, "Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory", Appl. Math. Model., vol. 38, no. 24, pp. 5934-5955, 2014.
[http://dx.doi.org/10.1016/j.apm.2014.03.036]
[26]
E.M. Miandoab, H.N. Pishkenari, A. Yousefi-Koma, and H. Hoorzad, "Polysilicon nano-beam model based on modified couple stress and Eringen’s nonlocal elasticity theories", Physica E, vol. 63, pp. 223-228, 2014.
[http://dx.doi.org/10.1016/j.physe.2014.05.025]
[27]
H. Matouk, "Investigation on hygro-thermal vibration of P-FG and symmetric S-FG nanobeam using integral Timoshenko beam theory", Adv. Nano Res., vol. 8, no. 4, pp. 293-305, 2020.
[28]
H-S. Shen, "Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells", Compos., Part B Eng., vol. 43, no. 3, pp. 1030-1038, 2012.
[http://dx.doi.org/10.1016/j.compositesb.2011.10.004]
[29]
H-S. Shen, "Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells", Compos. Struct., vol. 93, no. 8, pp. 2096-2108, 2011.
[http://dx.doi.org/10.1016/j.compstruct.2011.02.011]
[30]
K. Liew, Z. Lei, J. Yu, and L. Zhang, "Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach", Comput. Methods Appl. Mech. Eng., vol. 268, pp. 1-17, 2014.
[http://dx.doi.org/10.1016/j.cma.2013.09.001]
[31]
R. Ansari, and J. Torabi, "Numerical study on the buckling and vibration of functionally graded carbon nanotube-reinforced composite conical shells under axial loading", Compos., Part B Eng., vol. 95, pp. 196-208, 2016.
[http://dx.doi.org/10.1016/j.compositesb.2016.03.080]
[32]
SafarPour, "H., Ghanbari. B, and Ghadiri. M, “Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell", Appl. Math. Model., vol. 65, pp. 428-442, 2019.
[http://dx.doi.org/10.1016/j.apm.2018.08.028]
[33]
Y. Kiani, R. Dimitri, and F. Tornabene, "Free vibration of FG-CNT reinforced composite skew cylindrical shells using the Chebyshev-Ritz formulation", Compos., Part B Eng., vol. 147, pp. 169-177, 2018.
[http://dx.doi.org/10.1016/j.compositesb.2018.04.028]
[34]
M.H. Dindarloo, and L. Li, "Vibration analysis of carbon nanotubes reinforced isotropic doubly-curved nanoshells using nonlocal elasticity theory based on a new higher order shear deformation theory", In: Compos., Part B Eng., vol. 175. . 2019.107170
[http://dx.doi.org/10.1016/j.compositesb.2019.107170]
[35]
S. Sahmani, and A. Fattahi, "Nonlocal size dependency in nonlinear instability of axially loaded exponential shear deformable FG-CNT reinforced nanoshells under heat conduction", Eur. Phys. J. Plus, vol. 132, no. 5, p. 231, 2017.
[http://dx.doi.org/10.1140/epjp/i2017-11497-5]
[36]
A. Fattahi, and S. Sahmani, "Nonlocal temperature-dependent postbuckling behavior of FG-CNT reinforced nanoshells under hydrostatic pressure combined with heat conduction", Microsyst. Technol., vol. 23, no. 10, pp. 5121-5137, 2017.
[http://dx.doi.org/10.1007/s00542-017-3377-x]
[37]
H-S. Shen, "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., vol. 91, no. 1, pp. 9-19, 2009.
[http://dx.doi.org/10.1016/j.compstruct.2009.04.026]
[38]
Y. Han, and J. Elliott, "Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput. Mater. Sci., vol. 39, no. 2, pp. 315-323, 2007.
[http://dx.doi.org/10.1016/j.commatsci.2006.06.011]
[39]
A. Wang, H. Chen, Y. Hao, and W. Zhang, "Vibration and bending behavior of functionally graded nanocomposite doubly-curved shallow shells reinforced by graphene nanoplatelets", Results Phy., vol. 9, pp. 550-559, 2018.
[http://dx.doi.org/10.1016/j.rinp.2018.02.062]
[40]
B. Karami, and D. Shahsavari, "On the forced resonant vibration analysis of functionally graded polymer composite doubly-curved nanoshells reinforced with graphene-nanoplatelets", In: Comput. Methods Appl. Mech. Eng., vol. 359. . 2020.112767
[http://dx.doi.org/10.1016/j.cma.2019.112767]
[41]
A.C. Eringen, "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., vol. 54, no. 9, pp. 4703-4710, 1983.
[http://dx.doi.org/10.1063/1.332803]
[42]
A.C. Eringen, Nonlocal Continuum Field Theories., USA: Springer Science & Business Media, 2002.
[43]
H. Askes, and E.C. Aifantis, "Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results", Int. J. Solids Struct., vol. 48, no. 13, pp. 1962-1990, 2011.
[http://dx.doi.org/10.1016/j.ijsolstr.2011.03.006]
[44]
P. Phung-Van, Q.X. Lieu, H. Nguyen-Xuan, and M.A. Wahab, "Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates", Compos. Struct., vol. 166, pp. 120-135, 2017.
[http://dx.doi.org/10.1016/j.compstruct.2017.01.049]
[45]
B. Karami, and M. Janghorban, "On the dynamics of porous nanotubes with variable material properties and variable thickness", Int. J. Eng. Sci., vol. 136, pp. 53-66, 2019.
[http://dx.doi.org/10.1016/j.ijengsci.2019.01.002]